The purpose of this study is to test the safety of a combination of study drugs called selinexor (KPT-330) and ixazomib (MLN9708) with a low dose of dexamethasone. This study tests different doses of the study drug, selinexor to see which dose is safer in people. Depending on which group (dose level) you are in, the dose of selinexor will vary, but the ixazomib and dexamethasone doses will be the same among the groups.

Inclusion Criteria:

• MSKCC confirmed diagnosis of multiple myeloma that has relapsed or is resistant after therapy with at least one immunomodulatory drug (i.e. lenalidomide, thalidomide) and at least one proteasome inhibitor.

• Measurable multiple myeloma disease, defined as meeting at least 1 of the following criteria within 14 days prior to registration:

  1. A monoclonal Ig (M-protein) concentration on serum protein electrophoresis (SPEP) of ≥ 0.5 g/dL.
  2. Measurable urinary light chain secretion by quantitative analysis using urine protein electrophoresis (UPEP) of ≥ 200 mg/24 hours.
  3. Involved serum free light chain (FLC) level ≥ 10 mg/dL, provided the serum FLC ratio is abnormal.
  4. Presence of extramedullary plasmacytomas.

• Patient must be ≥ 18 years of age at time of informed consent.

• Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 within 14 days prior to registration.

• Adequate hematology laboratory values within 14 days prior to registration:

  • Neutrophils ≥ 1.0 × 10^9/L (growth factor support is not allowed).
  • Platelet count ≥75x10^9/L. Platelet support is permitted within 14 days although platelet transfusions to help participants meet eligibility criteria are not allowed within 72 hours (3 days) prior to the blood sample to confirm protocol eligibility.

• Adequate hepatic and renal function laboratory values within 14 days prior to registration:

  1. Aspartate aminotransferase (AST) and/or alanine aminotransferase (ALT) ≤ 3 × the upper limit of normal (ULN).
  2. Total bilirubin ≤ 1.5 mg/dL except for patients with a history of elevated total bilirubin, such as in Gilbert's.
  3. Calculated (Cockcroft and Gault formula) or measured creatinine clearance ≥ 30mL/min.

• Left ventricular ejection fraction (LVEF) ≥ 40% within 28 days prior to registration, evaluated by 2-D transthoracic echocardiogram (ECHO) or, if ECHO is not available, by multi-gated acquisition (MUGA) scan.

• Female patients who:

  • Are postmenopausal for at least 1 year before the screening visit, OR
  • Are surgically sterile, OR
  • If they are of childbearing potential, agree to practice 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug, OR
  • Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence [e.g., calendar, ovulation, symptothermal, post-ovulation methods] and withdrawal are not acceptable methods of contraception.)If patient is female and of childbearing potential, she must have a negative serum beta human chorionic gonadotropin (β-HCG) test within 14 days prior to registration and consent to ongoing pregnancy testing during the course of the study.

• Male patients, even if surgically sterilized (i.e., status post-vasectomy), must agree to one of the following:

  • Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug, OR
  • Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods] and withdrawal are not acceptable methods of contraception.)

For both male and female patients, effective methods of contraception must be used throughout the study and for 3 months following the last dose of study treatment.

• Willingness and ability to comply with scheduled visits, treatment plan, laboratory tests and other study procedures.
• Written informed consent in accordance with federal, local, and institutional guidelines. The patients must provide informed consent prior to the first screening procedure.

Exclusion Criteria:

• Female patients who are lactating or have a positive serum pregnancy test during the screening period.
• Failure to have fully recovered from the reversible effects of prior anti-cancer therapy.
• Major surgery within 14 days prior to the first dose of study drug.
• Radiotherapy within 14 days prior to the first dose of study drug. If the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of the first dose of the study medications.
• Disease-related central nervous system involvement.
• The subject has uncontrolled significant intercurrent illness including, but not limited to, ongoing or active infection, uncontrolled congestive heart failure, unstable angina pectoris within 6 months, stroke within 6 months, myocardial infarction within 6 months, or uncontrolled cardiac arrhythmias, uncontrolled hypertension.
• Active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive.
• Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol.
• Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent.
• Known GI disease or GI procedure that could interfere with the oral absorption of study medication including difficulty swallowing.
• Concurrent malignancy except for treated non-melanoma skin cancer, cervical carcinoma in situ and low-risk prostate CA being monitored without treatment.
• Patient has Grade 3 peripheral neuropathy or Grade 2 with pain on clinical examination during the screening period.
• Chemotherapy within 14 days of the start of this trial.
• Prior exposure to a SINE compound.
• Patients unwilling to comply with the protocol.